Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > isfin3-2 | Unicode version |
Description: Weakly Dedekind-infinite sets are exactly those which can be mapped onto . (Contributed by Stefan O'Rear, 6-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.) |
Ref | Expression |
---|---|
isfin3-2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfin32i 8766 | . 2 | |
2 | isf33lem 8767 | . . 3 | |
3 | 2 | isf32lem12 8765 | . 2 |
4 | 1, 3 | impbid2 204 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -. wn 3 -> wi 4
<-> wb 184 e. wcel 1818 class class class wbr 4452
com 6700
cwdom 8004 cfin3 8682 |
This theorem is referenced by: fin1a2lem7 8807 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-rep 4563 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-ral 2812 df-rex 2813 df-reu 2814 df-rmo 2815 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3435 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-pss 3491 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-tp 4034 df-op 4036 df-uni 4250 df-int 4287 df-iun 4332 df-br 4453 df-opab 4511 df-mpt 4512 df-tr 4546 df-eprel 4796 df-id 4800 df-po 4805 df-so 4806 df-fr 4843 df-se 4844 df-we 4845 df-ord 4886 df-on 4887 df-lim 4888 df-suc 4889 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-iota 5556 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600 df-fv 5601 df-isom 5602 df-riota 6257 df-ov 6299 df-oprab 6300 df-mpt2 6301 df-om 6701 df-1st 6800 df-2nd 6801 df-recs 7061 df-rdg 7095 df-seqom 7132 df-1o 7149 df-oadd 7153 df-er 7330 df-map 7441 df-en 7537 df-dom 7538 df-sdom 7539 df-fin 7540 df-wdom 8006 df-card 8341 df-fin4 8688 df-fin3 8689 |
Copyright terms: Public domain | W3C validator |